# Math 1551 Honors Calculus

### Fall 2009 Section 4

Last updated October 23, 2009.

### Professor James J. Madden

IMPORTANT INFORMATION:
Lunch lecture on November 3. Poster

Syllabus: Policies | Content

Schedule of lecture topics, links to notes and resources and homework assignments.

 Day and Date Section Topics Notes Homework Problems Assigned M 8/24 1.1 Rational and real numbers Quiz. Definitions. Algorithms for converting rationals from decimal to fraction notation and vice versa. Notes Convert 1/17 to decimal and convert 365.17365 to fraction. T 8/25 1.1 Absolute value Notes 10: 15, 18, 21, 23, 24 W 8/26 1.2 Linear functions Enumerating the rationals. Needs Mathematica; get student verion here. 19: 7, 8, 22, 24. Challenge Problem Th 8/27 -- Motion graphs, average rate of change. Also showed non-denumerability of irrationals Problem sheet F 8/28 2.1 Instantaneous rate of change approximated by averages. Reviewed equations of lines, and built on this. 58: 17, 66: 5 M 8/31 2.1, 2.2 Review, instananeous rate of change Sample test Sample test: "Ave. Rate of Ch." 3 T 9/1 2.8 Limits Formal definition; problem p. 115: 33. Limits at infinity. Read 2.8. Sample test: Limits,1,2. W 9/2 2.2 More on limits Examples of how a limit may fail to exist: break in graph, asymptote, oscillation. 77: 1-25 (odd) Th 9/3 2.4 limits (cont.) Continuity defined. Using "local equality" to compute limits. Factoring A^n -B^n. Prepare for test. Help session tonight in Prescott 212, 7PM. F 9/4 -- Test Get a copy of the test: with answers | without answers. Hand in corrected test on Friday, 9/11. M 9/7 -- Labor Day Vacation -- -- T 9/8 2.3 Review of test; limit rules. Proofs of the limit rules--if you're interested--can be found on page A18 (Appendix D). 82: 5, 10, 15, 20, 25, 30, 35, 40 W 9/8 2.6 Trig functions; Squeeze Thm. Activity guide HW Th 9/10 2.6 Trig functions (cont.) Do parts C. and D. of the Notes. F 9/11 2.7 Intermediate Value Theorem (IVT) M 9/14 -- Proof of IVT Notes on the theorems of calculus Thoughts about proof Ham Sandwich Theorem (page 107-8) T 9/15 3.1 Definition of derivative. (Deduced the power rule.) 125: 29, 30, 32, 34, 37, 39 W 9/16 3.2 Derivatives as functions. Leibniz notation. Linearity of differentiation. 139: 1, 3, 5, 7, 9 Th 9/17 3.2 Exponential function F 9/18 3.3 Product and quotient rule Deriving the quotient rule from the product rule and the reciprocal rule; examples. 148: 5, 10, 15, 20, 25, 30, 35 M 9/21 3.3 Deriving the reciprocal rule More examples 148: 40, 45 T 9/22 3.4 Applications; estimation Problem: if the surface area of a cube is 600 sq. ft. and it increase by 1 sq. ft., what is the increase in volume. Solved algebraically and by estimation with tangent line. Estimate the cube root of 1001. W 9/23 3.5 Higher derivatives Formulae for d^n y/ dx^n; acceleration; pictures 165: 5, 9, 11, 25, 27, 33, 35, 39, 40 Th 9/24 3.6, 3.7 Trig. functions. Chain Rule. Proof of chain rule. Leibniz notation versus functional notation. Examples. 170: 9, 11, 15, 32, 37. 178: 5, 11, 13, 21, 22. F 9/25 3.7 Chain Rule Activities with differential equations M 9/28 3.8 Implicit differentiation 185: 11, 19, 31 T 9/29 QUIZ. Inverse functions. "Money is the eats of all roovels." Suppose f and g are inverses and let y = f(x). 1a) Differentiate the equation f(g(x)) = x with respect to x. Then use the result to express g' in terms of f' and g. (See page 187). 1b) If y = f(x), then f'(a) = dy/dx evaluated when x = a. Similarly, g'(b) = 1/(dy/dx) evaluated when WHAT = b? 1c) Does it make sense to say that dx/dy is the derivative of g? 2) If g is the inverse of the exponential function, what is g'? (For help, see p. 193.) W 9/30 3.9 Derivatives of inverse trig functions 191: 11, 13 19, 23. 197: 4, 8, 9, 10. Th 10/1 Fall Holiday -- -- F 10/2 Fall Holiday -- -- M 10/5 3.9 Practice quiz and discussion 209: 61, 63, 65, 67, 69 T 10/6 3.11 Quiz on derivatives. Related rates. Ladder example discussed in detail. 205: 9, 14, 17. CHALLENGE: any one of 44, 45, 46. W 10/7 Ch. 3 Review. Th 10/8 Review F 10/9 -- TEST on Chapter 3. M 10/12 4.1 Approximation Using e^x to approxiamte 1+x in the Birthday Problem and SS# problem. (Aside on history of logic (Russell, Wittgenstein, Goedel, Turing, von Neuman) T 10/13 4.1, 4.2 Approximation (cont.); Least Upper Bounds Review of SS# problem. Discussion of LUB. A. Suppose f(N, k) = e^((-(k-1)(k) / (2N)), N = 1,000,000,000. Find the largest k such that f(N, k) < 1/2. B. Suppose X is a nonempty set of real numbers that has an upper bound. Let U be the set of upper bounds of X. Must U have a least element? (The answer is discussed here.) W 10/14 4.2 Maximum Value Theorem, Maxima on closed intervals Optional notes showing how to derive the MVT from the least upper bound principle here. (These note will almost certainly be incomprehensibe to you. If they interest you and you want to understand them, I'll explain them to you in my office.) Hand these in Friday, 10/16 -> 228: 25, 29, 33, 37, 41, 45 Th 10/15 4.2 Rolle's Theorem The lecture concentrated on the idea that mathematics is both a system of notation and techniques for representing and solving problems AND a structured, logical domain of knowledge. I drew a chart on the board showing how many of the concepts we have been discussing come together in Rolle's Theorem. 230: 87, 88 F 10/16 4.3 Mean Value Theorem Careful proof of MVT, based on Rolle. M 10/19 4.3 Increasing and decreasing functions T 10/20 4.3 237: 27, 29, 31, 39, 49 W 10/21 4.4 Concavity 244: 37, 41, 44, 50 Th 10/22 4.5 End behavior of polynomials Large numbers (names) Suppose f(x) = a_n x^n + ...+ a_0 is a polynomial of degree n with positive leading coefficient. Let M > 0. Find x_0 so that x > x_0 implies f(x) > M. F 10/23 4.5 Rational functions and asymptotes 258: 79, 81, 83, 87, 91. Complete work on problem from yesterday. M 10/26 4.6 Applied Optimization Agenda for the week: we'll have optimization problems in homework every day this week, and will review previous night's work at the beginning of each class; other new topics will be introduced as noted. 266-7: 11, 24, 25. Work sheet. T 10/27 4.6 More optimization Examined HW. Suppose C = a x + b y and A = x y, where a and b are positive constants. Maximize A if C is fixed. Minimize C if A is fixed. W 10/28 4.7 l'Hopital's rule 267: 36, 50. 277: 1, 5, 9, 13, 15, 19, 21. Th 10/29 4.8 Newton's Method 266: 16. 278: 41, 43, 45, 47, 49. 283: 5. Investigate f(x) = x^(x^(x^...))) and its inverse function g(x) = x^(1/x). F 10/30 4.9 Antiderivatives M 11/2 T 11/3 W 11/4 Chapter 4 test Th 11/5 F 11/6 M 11/9 T 11/10